for this workshop
Arithmetic reflection groups and crystallographic packings
American Institute of Mathematics, San Jose, California
Daniel Allcock, Alex Kontorovich, and Alice Mark
This workshop, sponsored by AIM and the NSF, will be devoted to the complete classification of maximal hyperbolic arithmetic reflection groups. About 15 years ago, the number of such was shown to be finite, leaving open the possibility of fully determining all such, in a way that makes them accessible for other applications, e.g., the study of crystallographic sphere packings. Recent advances, both theoretical and algorithmic, make it plausible that with sufficient collaborative effort by experts in diverse areas from geometry/topology, dynamics, arithmetic groups, and number theory, will be able to complete this program. In the 2020 online version of this workshop, three groups made progress formulating and resolving questions. In particular, the question of when an Apollonian-type group preserves a quadratic form was answered, and bounds were found on the parameters that can be used to determine the signature of that form.
The main topics of the workshop are:
- Restricting the search space by, on one hand combinatorial and geometric methods, and on the other, spectral gap (Ramanujan) methods
- Computation methods in arithmetic groups
- Vinberg-type and related algorithms
This event will be run as an AIM-style workshop. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.
The deadline to apply for support to participate in this workshop has passed.
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